potentials in cooperative tu-games

—

Mathematical Social Sciences 34 (1997) 175–190

Potentials in cooperative TU-games

*Emilio Calvo , Juan Carlos SantosEkonomia Aplikatauren Saila I, Departamento de Economia Aplicada I, Universidad del Pais Vasco, Avda.

Lehendakari Agirre, 83, 48015 Bilbao, Spain

Received 31 May 1996; received in revised form 30 April 1997; accepted 31 May 1997

Abstract

This paper is devoted to the study of solutions for cooperative TU-games which admit apotential function, such as the potential associated with the Shapley value (introduced by Hart andMas-Colell). We consider the finite case and the finite type continuum. Several characterizations ofthis family are offered and, as a main result, it is shown that each of these solutions can beobtained by applying the Shapley value to an appropriately modified game. We also study therelationship of the potential with the noncooperative potential games, introduced by Monderer andShapley, for the multilinear case in the continuum finite type setting. 1997 Elsevier ScienceB.V.

Keywords: Potential; Shapley value; Cooperative TU-games; Cost allocation; Potential games;Multilinear extension

1. Introduction

The potential approach is a successful tool in physics. The idea that a force can bederived by a potential and the term ‘‘potential function’’ were first introduced by DanielBernoulli (1738) in Hydrodynamics (see Kline, 1972). An illustrative example is thegravitational field. The gravitational force which acts on a particle is a function of thisposition in the space: f 5 f(r) 5 f(x, y, z). The work done, by moving a particlecontinuously from A to B through the path s, is

1

W 5E f(r) dr 5E f(s(t)) ? s9(t) dt,s 0

*Corresponding author. E-mail: [email protected].

0165-4896/97/$17.00 1997 Elsevier Science B.V. All rights reserved.PII S0165-4896( 97 )00015-2

176 E. Calvo, J.C. Santos / Mathematical Social Sciences 34 (1997) 175 –190

3where s : R→R , s(0)5A, s(1)5B and s is continuously differentiable. Thegravitational field is conservative in the sense that it is path independent i.e., W is the

3same for every path s from A to B. But a field is conservative if a function P:R →R

exists, such that

W 5E f(r) dr 5 2E =P dr 5 P(A) 2 P(B),s s

so that 2=P(r)5f(r).Hart, Mas-Colell (1989) were the first to introduce the potential approach in

cooperative transferable utility games. In a very remarkable result, they proved that theShapley value (see Shapley, 1953) can result as the vector of marginal contributions of aparticular potential function. The uniqueness of this function is implied by an efficientnormalization condition. Nevertheless, there are many other solutions which do notsatisfy efficiency, such as the semivalues, (see Dubey et al., 1981) but they can also beobtained by an associated potential. What we cover in this paper is a characterization ofthe family of solutions that admit a potential.

This paper is divided into six sections. Following this introduction, Section 2describes the basic model and gives the definition of the potential associated with asolution function. Section 3 characterizes the solutions that admit a potential on finitegames. We prove that every one of these solutions is the Shapley value of a modifiedgame. Section 4 extends the main results to the finite type continuum case and Section 5gives a relationship with the non-cooperative potential games. Finally, Section 6 showshow these results can be extended to solutions with weighted potentials.

2. Notation and definitions

The notation is as in Hart, Mas-Colell (1989). A game is a pair (N, v), where N,N isNthe finite set of players and v: 2 →R is the characteristic function, satisfying v(5)50. A

subset S,N is called a coalition, and v(S) is the worth of the coalition S. Given a game(N, v) and a coalition S,N, we write (S, v) for the subgame obtained by restricting v to

sS; that is, the domain of the function v is restricted to 2 . The space of all games isdenoted by G, and the space of all finitely additive games is denoted by FA, i.e.,

N N(N, v)[FA if v(S)5o v(i), for all S,N. We will denote also by G and FA the seti[S

of all games, and additive games, respectively, with finite player set N.

Definition 2.1. A solution c on G is a function c : G→FA which assigns to every gameN(N, v) exactly one element of FA .

Given a solution c on G and a game (N, v)[G, we write c(N, v) for the characteristicfunction of the additive game c(N, v) (and there will be no confusion); and for i[N the

inumber c(N, v)(i) will be denoted by c (N, v).iA solution c on G is said to be efficient if o c (N, v)5v(N), for all (N, v)[G. Onei[N

of the most well known efficient solutions is the Shapley value (Shapley, 1953) definedas:

E. Calvo, J.C. Santos / Mathematical Social Sciences 34 (1997) 175 –190 177

uSu! ? (uNu 2 uSu 2 1)!i ]]]]]]w (N, v) 5 O [v(S < i) 2 v(S)],uNu!S,N•i

for all i[N, and all (N, v)[G.We will use the definition of potential stated by Ortmann (1995a, definition 3.4):

Definition 2.2. A solution c on G admits a potential if there exists a function P : G→Rc

satisfying

iP (N, v) 2 P (N•i, v) 5 c (N, v),c c

for all (N, v)[G, N±5, and all i[N.The original definition of a potential of Hart and Mas-Colell includes an additional

condition of normalization:

P ([, v) 5 0, for ([, v) [ G.c

So, we say that a solution c on G admits a 0-normalized potential if it admits a potentialsatisfying the above 0-normalized condition.

The existence of a potential and a 0-normalized potential are equivalent, since thefunction

0P (N, v) 5 P (N, v) 2 P ([, v),c c c

is a 0-normalized potential if P is a potential. Furthermore, a solution c on G admits atc

most one 0-normalized potential.In Ortmann’s definition, the potential is unique up to an additive constant, and also

more similar to the original physical use of the potential.Solutions that admit a potential assign a scalar evaluation to each game in such a way

that a player’s payoff is his marginal contribution to this evaluation. The path-breakingresult of Hart, Mas-Colell (1989) can be stated as follows: a solution c on G is efficientand admits a potential if and only if c is the Shapley value on G.

3. Characterization

This section is devoted to characterizing the family of all solutions on G that admit apotential. The main theorem shows that any solution that admits a potential turns out tobe the Shapley value of an auxiliary game. This game is defined as follows:

Definition 3.1. Given a solution c on G and a game (N, v)[G, we define the auxiliarygame (N, v ) as:c

iv (S) 5O c (S, v), for all S , N.ci[S


Remark 3.2. Note that the worth of a coalition S in the game (N, v) does not depend ofNN, and hence, we denote the characteristic function by v and not, for example, v .c c

Note also, that if c satisfies efficiency then (N, v )5(N, v), for all (N, v)[G.c

Now, we are ready to state and prove the main theorem:

Theorem A. Let c be a solution on G. Then c admits a potential if and onlyif c(N, v) 5 c(N, v ), for all (N, v)[G.c

Proof. First suppose that c(N, v)5w(N, v ) for all (N, v)[G. Since we know that thec

Shapley value w has a 0-normalized potential P (the Hart and Mas-Colell potential), wew

can define a potential of c as:

P (N, v) 5 P (N, v ) for all (N, v) [ G.c w c

Then, for i[N (by remark 3.2):

i iP (N, v) 2 P (N•i, v) 5 P (N, v ) 2 P (N•i, v ) 5 w (N, v ) 5 c (N, v),c c w c w c c

and hence, P is a potential of c.c

Assume now that c has a potential P . To prove that c(N, v)5w(N, v ), we will usec c

an induction argument for the cardinal of N, uNu. If uNu51, suppose that N5hij. Then,ithe game (hij, v ) satisfies v (i)5c (hij, v), and hence, by the efficiency of the Shapleyc c

0value, c(hij, v)5w(hij, v ). Furthermore, the 0-normalized potential P of c satisfiesc c0P (hij, v)5P (hij, v ), where P is the 0-normalized potential of the Shapley value.c w c w

Now let (N, v)[G, uNu.1, i[N, and assume by the induction hypothesis that0 0

c(S, v)5w(S, v ) and P (S, v)5P (S, v ) holds for all S,N, S±N, where P and P arec c w c c w

the 0-normalized potentials. By definition of the potential, we have:

i i 0 0c (N, v) 2 w (N, v ) 5 P (N, v) 2 P (N•i, v) 2 P (N, v ) 1 P (N•i, v ),c c c w c w c

0 i ibut P (N\i, v)5P (N\i, v ) by induction; then c (N, v)2w (N, v )5d for all i[N. Usingc w c c

the efficiency of w and the definition of (N, v ) we have:c

i iuNu ? d 5O c (N, v) 2O w (N, v ) 5 v (N) 2 v (N) 5 0,c c ci[N i[N

i ihence c (N, v)5w (N, v ), for all i[N. jc

We will now give some examples of solutions that admit a potential. For example, thefamily of semivalues (see Dubey et al., 1981). There is a one to one mapping betweensemivalues and probability measures m on [0, 1] which allows us to give an explicitformula for its calculation:

1

i uS u uN•(S<i )uw (N, v) 5 O E t ? (1 2 t) dm(t) ? [v(S < i) 2 v(S)],m 3 4S,N•i

0

for all i[N, and (N, v)[G. It can be checked that the potential is:


1

uS u21 uN•S uP (N, v) 5 O E t ? (1 2 t) dm(t) ? v(S).wm 3 4S,N0S±[

The Shapley value is an element of this family obtained by using the Lebesguemeasure l on [0, 1], so the potential for the Shapley value is:

(uSu 2 1)! ? uN•Su!]]]]]P (N, v) 5 P (N, v) 5 O v(S).w wl uNu!S,N

S±[

which is the well known formula obtained by Hart, Mas-Colell (1989).

Remark 3.3. Several modifications for the value, due to some kind of restrictions orconstraints in the communication between players, have been proposed in the literature.For example, the game can be modified by a graph (Myerson, 1977), by a conferencestructure (Myerson, 1980), by a permission structure (Gilles et al., 1992), by arestriction (Derks, Peters, 1993) or by a probabilistic graph (Calvo et al., 1995). All ofthese values are the Shapley value of an appropriate modified game, so they fall into thefamily of solutions that have a potential (hint, take the potential associated to each oneof these solutions as the potential of its corresponding modified game).

Nevertheless, the class of solutions that admit a potential also includes peculiarsolutions such as the constant solutions. Given a function f : N→R, a constant solutionassociated to f is defined by:

ic (N, v) 5 f(i),f

for all (N, v)[G and i[N. The 0-normalized potential associated with each solution isdefined by

P (N, v) 5O f(i).cfi[N

Finally, note that there exist solutions that do not admit a potential, for example, thenucleolus.

We conclude this section showing the close relation between the potentializability of asolution and the properties of balanced contributions and path independence.

Balanced contributions was a property introduced by Myerson (1980). This principleasserts that, for any two players, the amount that each player would gain or lose by theother’s withdrawal from the game, should be equal. Myerson used this principle toextend the Shapley value to games with conference structures. However, this result givesan axiomatization for the Shapley value in the special case of TU-games.

Formally, a solution c on G satisfies balanced contributions if

i i j jc (N, v) 2 c (N•i, v) 5 c (N, v) 2 c (N• j, v),

for all (N, v)[G and hi, jj[N.


The Myerson result is that a solution c on G is efficient and satisfies balancedcontributions if and only if c is the Shapley value on G.

In Ortmann, (1995a, proposition 3.5) this result is extended in order to characterize allthe solutions that admit a potential by means of balanced contributions.

The characterization of the potential in physics by means of path independence has itsanalogue in cooperative TU-games. To see this, take a game (N, v)[G and suppose thatwe want to buy off the players one by one, in order to leave the game. For that, starting

iwith player i[N we pay him c (N, v) and then i leaves the game. Now, we choosejanother player j[N\i and pay him c (N\i, v), and so on. Solution c will be path

independent if, according to this rule, the sum of the payoffs to buy off the set of playersN, is the same for every order in which the players are chosen.

Formally, given a finite set of players N, an order on N is a one to one function v :N→N. For all i[N, v(i) is the position of player i in the order v. Let V(N) be the set

vof all orders on N and let R be the set of all predecessors of i in order v, i.e.,i

vR 5 h j [ N: v( j) , v(i)j.t

A solution c on G satisfies path independence if for all (N, v)[G. and all v,v9[V(N), it holds that:

i v i v 9O c (R < i, v) 5O c (R < i, v).i ii[N i[N

In Hart, Mas-Colell (1992) it is proved that the Shapley value satisfies this property,and Ortmann, (1995a, proposition 3.10) extends also this result for all the solutions thatadmit a potential. In Monderer, Shapley (1996), non-cooperative games which admit apotential are studied (see Section 5 of this work for more details). These games arecharacterized (theorem 2.8) by using also a property of path independence, which is thesame standard approach to potential functions as used in physics.

Following Ortmann’s results and theorem A in this paper it is straightforward to provethe next result:

Corollary 3.4. Let c be a solution on G. The following are equivalent:

(i) c admits a potential.

(ii) c satisfies the balanced contributions axiom.

(iii) c satisfies path independence.

(iv) c(N, v)5w(N, v ), for all (N, v)[G.c

Note that by remark 3.2 c is efficient if and only if (N, v)5(N, v ), for all (N, v)[G,c

and the equivalence of (i) and (iv) yields a new proof of the Hart and Mas-Colelltheorem.


4. The finite type continuum case

The nearest setting in game theory to physics are the TU games with a finite typecontinuum of players, as, for example, the cost allocation problems (see Billera, Heath,

n1982; Mirman, Tauman, 1982). In this context, y5( y , . . . ,y )[R represents a1 n 1

coalition and y is the number (or ‘‘mass’’) of players of type i. A game is given by aincharacteristic function f : R →R, with f(0)50. A type-symmetric solution for f is a1

n n ifunction c( f, ?): R →R , where c ( f, y), y..0, represents the per capita payoff of11iplayers of type i, so y ?c ( f, y) is the total value that type i receives. The game f is fixedi

in this section.nLet D be the set of all solutions c( f, ?) continuously differentiable on R and with11

n a ilim o y ?c ( f, y)50 for some a ,1. The last condition can be seen as ay→0 i51 i

condition of null players, in the sense that the total value tends to 0 when y tends to 0.For example, solutions that are homogeneous of degree a, with a .21, are included

]]in this family. They can be associated to cost functions, such as y ? y (see Hart,œ 1 21Mas-Colell (1995a) for a more detailed illustration). In Ortmann (1995b) the potential

approach is also considered for games with a continuum of players of finitely manytypes, but in his slightly more restrictive context, the above cost functions are not

2included in the domain. This require the following definition :

1Definition 4.1. A solution c( f, ?) admits a potential if there exists a C functionnP ( f, ?): R →R satisfying:c 11

n1. =P ( f, y)5c( f, y) for all y[R ,c 11n2. lim P ( f, ty) exists and is finite for all y[R ,t→01 c 11

iwhere = P ( f, y)5(≠ /≠x )P ( f, y).c i c

Remark 4.2. As in Hart, Mas-Colell (1989) we can define a normalized potential as1follows: a solution c( f,?) admits a 0-normalized potential if there exists a C function

nP ( f, ?): R →R, satisfying:c 11

n1. =P ( f, y)5c( f, y) for all y[Rc 11n2. lim P ( f, ty)50, for all y[R .t→01 c 11

The existence of a potential and a 0-normalized potential are equivalent. Obviously,every 0-normalized potential is a potential. Conversely. the function

0P ( f, ? ) 5 P ( f, ? ) 2 lim P ( f, ty) ,F Gc c ct→01

is a 0-normalized potential if P is a potential.c

1In Hart, Mas-Colell (1995a), (1995b) the potential approach is extended to the non transferable utility case forthe continuum finite type setting.

2To include these interesting solutions, this definition is less restrictive than that proposed by Ortmann, (1995b,point (ii) in theorem 1.4).


A continuously differentiable solution c( f, ?) satisfies the balanced contributionsaxiom if

≠ ≠i j] ]c ( f, y) 5 c ( f, y),≠x ≠xj i

nfor all y[R and all i, j[h1, . . . ,nj.11

The next theorem makes it possible to extend proposition 3.5 in Ortmann (1995a) togames with a finite type continuum of players.

Theorem B. Let c( f, ?)[D be a solution. Then c( f, ?) admits a potential if and only ifc( f, ?) satisfies the balanced contributions axiom.

Proof. First we prove the following result:

Lemma 4.3. If c( f, ?)[D it holds that:1i) the integral e y?c( f, ty)dt exists, for all y..0.0

iii) lim t?c ( f, ty)50, for all y..0 and i[h1, . . . ,nj.t→01

Proof. Let c( f, ?)[D. Then

1 112an yi a a i]]E y ? c( f, ty) dt 5E O ? ( y ? t ) ? c ( f, ty) dt, for all y 4 0,a iti51

0 0

1and thus, the integral e y?c( f, ty)dt exists. Now, to prove (ii) note that0i ilim t?c ( f, ty)5lim (1/y )ty ?c ( f, ty), and the last expression tends to be zerot→01 t→01 i i

when c( f, ?)[D. j

Proof of theorem B. Let c( f, ?)[D be a solution that admits a potential P ( f, ?). Then,c

for every y..0 we have:

≠ ≠ ≠ ≠ ≠ ≠i j] ] ] ] ] ]c ( f, y) 5 P ( f, y) 5 P ( f, y) 5 c ( f, y),S Dc S c D≠x ≠x ≠x ≠x ≠x ≠xj j i i j i

and hence, the balanced contributions axiom holds.Now, let c( f, ?)[D be a solution and suppose that c( f, ?) satisfies the balanced

1contributions axiom. We define P ( f, y)5e y ?c( f, ty)dt, where y..0. By (i) ofc 0

lemma 4.3, P ( f, y) is well defined and by (ii), lim P ( f, ty)50. Moreover, by (ii)c t→01 c

of the same lemma, for every y..0, it follows that:

1

di i i i]c ( f, y) 5 c ( f, y) 2 lim ´ ? c ( f, ´y) 5E (tc ( f, ty)) dt.dt´→01

0

Then


1 1

d di i i i] ]S Dc ( f, y) 5E (tc ( f, ty)) dt 5E c ( f, ty) 1 t (c ( f, ty)) dtdt dt0 0

1n

≠i i]5E c ( f, ty) 1O ty (c ( f, ty)) dt.S Dj ≠xjj510

As c( f, ?) satisfies the balanced contributions axiom, we have:

1 1n

≠ ≠i i i] ]c ( f, y) 5E c ( f, ty) 1O ty (c ( f, ty)) dt 5E ( y ? c( f, ty)) dtS Dj ≠x ≠xj ij510 0

≠]5 P ( f, y),c≠xi

where the last equality follows from theorems for derivatives of the improper integrals.Then P ( f, ?) is the 0-normalized potential of c( f, ?) and the result is proved. jc

nA solution c( f, ?) is efficient if y?c( f, y)5f( y), for all y[R . Hart, Mas-Colell11

(1989) proved that there is only one solution that admits a potential and satisfiesefficiency; this is the well known Aumann–Shapley prices solution (see Billera, Heath,1982). This solution w( f, ?) is defined as:

1

≠fi n]w ( f, y) 5E (ty) dt, for all y [ R and i [ N.11≠xi0

Hart, Mas-Colell (1989) give an explicit formula for the potential of w as follows:

1

1 n]P ( f, y) 5E ? f(ty) dt, for all y [ R .w 11t0

The relationship between the solutions that have a potential and the Aumann–Shapleyprices solution is given in the next theorem that extends theorem A to games with afinite type continuum of players. First, we define an auxiliary game in this context.

Given a solution c( f, ?)[D, we define an auxiliary continuum game f as follows:c

nf ( y) 5 y ? c( f, y) for all y [ R .c 11

Theorem 4.4. Let c( f, ?)[D be a solution. Then c( f, ?) admits a potential P if andcnonly if c( f, y)5w( f , y), for all y[R .c 11

Proof. Since the Aumann–Shapley prices admit a potential, one direction is immediate.For the other direction, suppose that c( f, ?)[D has a potential P ( f, ?). Then, fromc

theorem B it follows that:


1 1

≠ ≠ ≠ 1i ] ] ] ]S Dc ( f, y) 5 P ( f, y) 5E ( y ? c( f, ty)) dt 5E f (ty) dtc c≠x ≠x ≠x ti i i0 0

≠ i]5 P ( f , y) 5 w ( f , y),w c c≠xi

for all y..0, and then, the proof is complete. j

Note that this theorem yields a simple proof of Ortmann’s nonatomic analogue of theHart and Mas-Colell theorem for finite type continuum games (theorem 1.4 in Ortmann,1995b) since c is efficient if and only if f 5f.c

5. Potential games

An interesting subfamily of non-cooperative games are the so called Potential games.Rosenthal (1973) was the first to use potential functions for games in a strategic form,although Monderer, Shapley (1996) were the first to introduce these games explicitly,giving an extensive study of their properties (see also Neyman, 1991; Qin, 1992;Milchtaich, 1993 and Dutta et al., 1995 for some applications). Monderer and Shapleyillustrated in their paper that it is possible to give a strategic approach to value theory byusing special non-cooperative games, known as participation games. More specifically,they proved that when the Shapley value is used to determine the payoff functions, thisparticipation game is a potential game. To prove it, they use the fact that the Shapleyvalue can be obtained by using the Hart and Mas-Colell potential function.

1 nLet G 5G(u , . . . ,u ) be a game in strategic form with a finite number of players:iN5h1, . . . ,nj. The set of strategies for each player i is Y , and the payoff function of

i iplayer i is u : Y→R, being the set of strategy profiles Y53 Y . A function P: Y→Ri[Njis a potential for G, if for every i[N and X [3 Y , it follows that:N \i j[N \i

i iu (x , x ) 2 u (x , z ) 5 P(x , x ) 2 P(x , z )N•i i N•i i N•i i N•i i

i 3for all x , z [Y . A potential game is a non-cooperative game that admits a potential.i iNNow let’s take a finite cooperative game v[G and for every solution c a

participation game in the strategic form G(c, v) is defined as follows:iThe set of players is N. The set of strategies for player i is Y 5h0, 1j. If player i

chooses 0 he decides not to join the game and receives a payoff of 0; if he chooses 1 hedecides to participate in the game. Let N(x) be the set of players where x 51. Then thei

ipayoff function u of player i is:

i iu (x) 5 c (v ).N(x)

In Monderer, Shapley (1996) the following result (see theorem 6.1) is proved:

3In Monderer, Shapley (1996) this function P is called an exact potential for G, but since is the only one usedhere, we call it simply by a potential for G.


NTheorem 5.1. Let c be an efficient solution and let v[G . Then c is the Shapley valueif and only if G(c, v) is a potential game.

Now we consider the continuum case. For this we will work with differentiablestrategic games. Assume that the strategy set for every player is an interval of real

inumbers, and suppose that the payoff functions u are continuously differentiable. Then(see lemma 4.4 in Monderer and Shapley) P is a potential for G if and only if P iscontinuously differentiable, and

i≠u ≠P] ]5 , for every i [ N.≠x ≠xi i

Moreover, if the payoff functions are twice continuously differentiable (see theorem4.5) G is a potential game if and only if

2 i 2 j≠ u ≠ u]] ]]5 , for every hi, jj , N.≠x ≠x ≠x ≠xi j i j

The next lemma about separability of the payoff function will be useful in the next4theorem (we omit the proof ):

Lemma 5.2. Let G be a potential game such that the payoff functions are twiceicontinuously differentiable and o u (x)50 for every x[Y, theni[N

i iju (x) 5O g (x ), for each i [ N.jj[N

nWe will say that a function f : R →R is multilinear if it is twice continuously2 2differentiable and (≠ f /(≠x ) )50 for every i[h1, . . . ,nj.i

Given a finite type continuum game f, with a set of player types N5h1, . . . ,nj and an nvector y[R , for each solution c on R we define a non-cooperative game G(c, f, y)1 1

in its strategic form as follows:The set of players is N5h1, . . . ,nj. The set of strategies of each player i is the interval

iY 5[0, y ] and the payoff function is defined by:i

i iu (x) 5 x ? c ( f, x).i

nTheorem 5.3. Given a multilinear game f and a vector y5R , let c be an efficient11

and a twice continuously differentiable solution, then c is the Aumann–Shapley pricessolution if and only if G(c, f, y) is a potential game.

Proof. Existence: let w be the Aumann–Shapley prices solution, then

4 2The result follows by simply taking into account that f(x, y)5g(x)1h( y) is implied by (≠ f /≠x ≠y)50.


1i i

≠u ≠w ≠ ≠fi i i] ] ]](x) 5 w ( f, x) 1 x ? ( f, x) 5 w ( f, x) 1 x ?E (tx) dt 5 w ( f, x),i i≠x ≠x ≠x ≠xi i i i0

since f is multilinear. Moreover, w has a potential P and then =P ( f, x)5w( f, x), sow w

P5P and G(w, f, y) is a potential game.w

Uniqueness: suppose that there are two different efficient solutions c and c such1 2

that G(c , f, y) and G(c , f, y) are potential games. Therefore, c 2c is an efficient1 2 1 2

solution for the null game g(x)50, and game G(c 2c , g, y) is a potential game. Then1 2i iall the hypotheses of lemma 5.2 are satisfied; hence u 2u is a separable function. i.e.,1 2

i i i j(u 2 u )(x) 5O g (x ), for every i [ N.1 2 jj[N

But, by definition

i i ijO g (x ) 5 x ? (c ( f, x) 2 c ( f, x)),j i 1 2j[N

i i i iand since u 2u is twice continuously differentiable, it means that u 2u is a function1 2 1 2

of x , i.e.,i

i i i(u 2 u )(x) 5 g (x ). (1)1 2 i

Using efficiency again, it follows that:

i i iO (u 2 u )(x) 5O g (x ) 5 0,1 2 ii[N i[N

which implies that:

i i j(u 2 u )(x) 5 2 O g (x ). (2)1 2 jj[N•i

But the only possibility for both (1) and (2) to be satisfied simultaneously is that

i i(u 2 u )(x) 5 0, for every i [ N,1 2

and then c 2c 50, and the proof is complete. j1 2

When f is not multilinear, the result is no longer true. Even if the game G(w, f, y) is apotential game, its potential, P, does not necessarily coincide with the Hart andMas-Colell potential associated with w, P . For example, if g:R→R is a twicew

continuously differentiable function and

f(x , . . . ,x ) 5 g P x 5 g(x ? ? ? x ),S D1 n i 1 ni[N

then, G(w, f, y) is a potential game, and its potential is1 1

≠f n21]P(x , . . . ,x ) 5 x ?E (tx) dt 5 x ? ? ? x ?E t ? g9(tx ? ? ? tx ) dt,1 n i 1 n 1 n≠xi0 0


for any i[N; but the Hart and Mas-Colell potential associated to w is1 1

1 1] ]P (x , . . . ,x ) 5E f(tx) dt 5E ? g(tx ? ? ? tx ) dt.w 1 n 1 nt t

0 0

6. Final remarks

6.1. Weights

This remark is devoted to extending the results of this paper to weighted finite andcontinuum problems. First, we study the finite case.

We will assume that a collection of weights is exogenously given and that we want toconsider solutions which balance the players according with those weights. Formally, asystem of weights is a function w:N→R, with w(i).0 for all i[N. We will denote

iw 5w(i). A solution c on G satisfies the w-balanced contributions axiom if

1 1i i j j] ](c (N, v) 2 c (N• j, v) 5 (c (N,v) 2 c (N•i, v)),i jw w

for all (N, v)[G and hi, jj,N.A solution c on G admits a w-potential if there exists a function P : G → Rcw

satisfying

i iw (P (N, v) 2 P (N•i, v)) 5 c (N, v),c cw w

for all (N, v)[G, N±5, and all i[N.The next theorem extends corollary 3.4 to the weighted case. The proof is equivalent

to the symmetric case and we omit it.

Theorem 6.1.1. Let c be a solution on G. The following are equivalent:

(i) c admits a w-potential.

(ii) c satisfies the w-balanced contributions axiom.

(iii) c(N, v)5w (N, v ), for all (N, v)[G, where w is the weighted Shapley value (seew c w

Kalai, Samet, 1988).In the continuum case we need only define w-potential and w-balanced contributions

since the other adaptations are straightforward. A weights system w is a function1w:h1, . . . ,nj→R . A solution c( f, ?) admits a w-potential if there exists a C function11

nP ( f, ?):R →R, satisfying:c 11w

n1. w ?=P ( f, y) 5 c( f, y), for all y[R ,c 11wn2. lim P ( f, ty) exists and is finite for all y[R ,c 11wt→01

iwhere = P ( f, y)5 (≠ /≠x ) P ( f, y)c i cw w

A continuously differentiable solution c( f, ?) satisfies the w-balanced contributionsaxiom if


1 ≠ 1 ≠i j]] ]]c ( f, y) 5 c ( f, y),i j≠x ≠xw wj i

nfor all y[R and all i, j[h1, . . . ,nj.11

Then the weighted version for theorem B is:

Theorem 6.1.2. Let c( f, ?)[D be a solution. Then c admits a w-potential if and only ifc satisfies the w-balanced contributions axiom.

6.2. Multilinear extensions

The multilinear extension of a game was a concept introduced by Owen (1972).NGiven a game v[G , the multilinear extension of v is a function f , of uNu real variablesv

5defined by:

f (x) 5 f (x , . . . ,x ) 5 O P x ? P (1 2 x ) ? v(S).F Gv v 1 n i ii[S i[N•SS,N

S±[

NIf we restrict the domain of f to take values in the unit cube [0, 1] , f can be seen asv v

the probabilistic continuum type version of finite game v. It is shown that by using themultilinear approach, the Shapley value of v can be obtained by integrating the gradientof f along the main diagonal of the cube, i.e.,v

1

≠fii ]w (v) 5E (t, . . . ,t) dt, for every i [ N.≠xi

0

This means that wv is the Aumann–Shapley prices solution of f evaluated atv

y5(1, . . . ,1), i.e., w(v)5w( f , 1). The same relationship can also be found betweenv

potentials; it is easy to check that1

1]P ( f , 1) ; P (v) 5E f (t ? 1) dtw v w vt

0

and

P (v) 2 P (v ) 5 P ( f , 1) 2 P ( f , 1 )w w N•i w v w v N•iN•i

1 1

≠f1 v] ]5E ( f (t ? 1) 2 f (t ? 1 )) dt 5E (t ? 1) dtv v N•iN•it ≠xi

0 0

≠]5 P ( f , 1).w v≠xi

5In Owen’s original definition, the rank of the summation includes the empty set as well, but v(5)50 so thatthe two definitions are equivalent.


Notice that these relationships remain true when replacing w by w, where m is am1nonnegative measure on ([0, 1], @). When e 1dm(t)51, we fall into the family of0

semivalues.Furthermore, all the solutions defined in the continuum type setup can be extended to

the family of finite games, by using the multilinear approach.

Acknowledgements

Research supported by grants from the Universidad del Pais Vasco (project UPV036.321-HA 012/94) and the Gobierno Vasco (project P195/101). We acknowledgeuseful discussions with Sergiu Hart, Andreu Mas-Colell and Abraham Neyman. Also,many helpful comments of an anonymous referee are gratefully acknowledged.

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potentials in cooperative tu-games

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